On the Fourier-Borel Transformation and Spaces of Entire Functions in a Normed Space

Abstract

Publisher Summary This chapter discusses the spaces of entire functions in a normed space that are of the order (nuclear order) k and type (nuclear type) strictly less than A . The corresponding spaces in which the type is allowed to be also equal to A are introduced. These spaces have natural topologies and they are infinite dimensional analogous of the spaces considered in Martineau. This chapter investigates the Fourier–Borel transformation in these spaces and presents that these transformations identify the algebraically and topologically strong duals of the spaces with other spaces of the same kind. The chapter also proves existence and approximation theorems for convolution equations in these spaces. The spaces of entire functions in normed spaces are discussed in the chapter. The chapter presents a characterization of various bounded subsets. These characterizations are used in the proof that the Fourier–Bore1 transformations are topological isomorphisms.